Simplifying and Solving: 5x = 1/2 + 3x

Q: Why do I subtract 3x instead of adding it?

You want to collect all x terms on one side. Since 5x is larger than 3x, subtract 3x from both sides to get 2x on the left and eliminate x from the right side.

Q: Can I subtract 5x from both sides instead?

Yes! You'd get $ 0 = \frac{1}{2} - 2x $, then solve for x. Both methods work, but it's usually easier to keep the positive coefficient for x.

Q: How do I divide a fraction by 2?

To divide $ \frac{1}{2} \div 2 $, multiply by the reciprocal: $ \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} $. Or think: half of a half is a quarter!

Q: What if I get confused about which side to put the variables?

It doesn't matter! You can collect variables on either side. Just be consistent and subtract the same term from both sides to maintain equality.

Q: Why is my answer a fraction instead of a whole number?

Not all equations have whole number solutions! Fractions are perfectly valid answers. The important thing is that your fraction satisfies the original equation when you check it.

Linear Equations with Variable Collection

Solve for x:

$5x=\frac{1}{2}+3x$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Arrange the equation so that X is isolated on one side

00:12 Combine like terms

00:16 Isolate X

00:30 Simplify where possible

00:33 Make sure to multiply numerator by numerator and denominator by denominator

00:37 This is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for x:

$5x=\frac{1}{2}+3x$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Subtract $3x$ from both sides of the equation.
Step 2: Simplify and solve for $x$ .

Now, let's work through each step:

Step 1: Subtract $3x$ from both sides
Starting with the equation $5x = \frac{1}{2} + 3x$ , subtract $3x$ from both sides to get:
$(5x - 3x) = \frac{1}{2}$ .

Step 2: Simplify and solve for $x$
Simplifying the left side yields $2x = \frac{1}{2}$ .
Next, divide both sides of the equation by 2 to isolate $x$ :
$x = \frac{1}{2} \div 2$ .

When dividing $\frac{1}{2}$ by 2, we are effectively finding half of $\frac{1}{2}$ , which is:
$x = \frac{1}{4}$ .

Therefore, the solution to the problem is $x = \frac{1}{4}$ .

Final Answer

$\frac{1}{4}$

Key Points to Remember

Essential concepts to master this topic

Isolation Rule: Collect all variable terms on one side first
Technique: Subtract 3x from both sides: 5x - 3x = 2x
Check: Substitute $x = \frac{1}{4}$ : $5(\frac{1}{4}) = \frac{1}{2} + 3(\frac{1}{4})$ gives $\frac{5}{4} = \frac{5}{4}$ ✓

Common Mistakes

Avoid these frequent errors

Adding 3x to both sides instead of subtracting
Don't add 3x to both sides = 8x = 1/2 + 6x, which gives x = -1/4! This moves variables in the wrong direction. Always subtract the smaller variable term from both sides to collect variables properly.

Practice Quiz

Test your knowledge with interactive questions

$$ -16+a=-17 $$

FAQ

Everything you need to know about this question

Why do I subtract 3x instead of adding it?

You want to collect all x terms on one side. Since 5x is larger than 3x, subtract 3x from both sides to get 2x on the left and eliminate x from the right side.

Can I subtract 5x from both sides instead?

Yes! You'd get $0 = \frac{1}{2} - 2x$ , then solve for x. Both methods work, but it's usually easier to keep the positive coefficient for x.

How do I divide a fraction by 2?

To divide $\frac{1}{2} \div 2$ , multiply by the reciprocal: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ . Or think: half of a half is a quarter!

What if I get confused about which side to put the variables?

It doesn't matter! You can collect variables on either side. Just be consistent and subtract the same term from both sides to maintain equality.

Why is my answer a fraction instead of a whole number?

Not all equations have whole number solutions! Fractions are perfectly valid answers. The important thing is that your fraction satisfies the original equation when you check it.

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